04win-m1sols

04win-m1sols - SOLUTIONS TO MATH 51 MIDTERM 1 January 29,...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: SOLUTIONS TO MATH 51 MIDTERM 1 January 29, 2004 1. Find all solutions of the following system: x 1- x 2 + x 3 + 2 x 4 = 3 x 2 + x 3 + x 4 = 3 x 1 + x 2 + 3 x 3 + 4 x 4 = 9 Solution. Write the augmented matrix and then use Gaussian elimination: 1- 1 1 2 3 1 1 1 3 1 1 3 4 9 1- 1 1 2 3 1 1 1 3 2 2 2 6 1 0 2 3 6 0 1 1 1 3 0 0 0 0 0 So x 1 + 2 x 3 + 3 x 4 = 6 x 2 + x 3 + x 4 = 3 The free variables are x 3 and x 4 , so the solutions are: x 1 = 6- 2 x 3- 3 x 4 , x 2 = 3- x 3- x 4 , ( x 3 R , x 4 R ) 2. Let L be the intersection of the two planes x + y + z = 4 and 2 x + 3 y + z = 9 . Find a parametric equation for L . Solution. Write the augmented matrix and use Gaussian elimination: fl fl fl fl 1 1 1 4 2 3 1 9 fl fl fl fl fl fl fl fl 1 1 1 4 0 1- 1 1 fl fl fl fl fl fl fl fl 1 0 2 3 0 1- 1 1 fl fl fl fl . so x + 2 z = 3 and y- z = 1, or (moving the free variable z to the right hand side) x = 3- 2 z and y = 1 + z . Thus the intersection is given by x y z = 3- 2 z 1 + z z = 3 1 + z - 2 1 1 . 3(a) . Suppose u , v , and w are points in R n such that k u k = k v k = k w k = 1 and such that w =- u . Suppose also that v is not equal to u or to w . Prove that the triangle uvw has a right angle at v . 1 Solution. The vector from u to v is v- u . The vector from w to v is v- w . We want to show that these two vectors are orthogonal, so we calculate the dot product: ( v- u ) ( v- w ) = ( v- u ) ( v + u ) = v v + v u- u v- u u = k v k 2-k u k 2 = 1 2- 1 2 = 0 3(b) . Suppose x , y , and z are vectors in R n whose norms are 1, 2, and 3, respec- tively. Suppose each vector is orthogonal (i.e., perpendicular) to each of the other two. Find a scalar c such that the vector x + c y- z is orthogonal to the vector x + y + z . Solution The vectors x + y + z and x + c y- z vectors will be orthogonal provided their dot product is 0. When we multiply it out (i.e., use the distributive property), all the mixed terms ( x y , x z , etc.) are 0 by orthogonality. So 0 = ( x + y + z ) (...
View Full Document

Page1 / 7

04win-m1sols - SOLUTIONS TO MATH 51 MIDTERM 1 January 29,...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online